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A354225
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Lexicographically earliest sequence of distinct positive integers such that a(1) = 1 and for any n > 1, n / gcd(n, a(n)) and a(n) / gcd(n, a(n)) are prime.
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1
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1, 3, 2, 6, 7, 4, 5, 12, 15, 14, 13, 8, 11, 10, 9, 24, 19, 27, 17, 28, 33, 26, 29, 16, 35, 22, 18, 20, 23, 42, 37, 48, 21, 38, 25, 54, 31, 34, 51, 56, 43, 30, 41, 52, 63, 58, 53, 32, 77, 70, 39, 44, 47, 36, 65, 40, 69, 46, 61, 84, 59, 74, 45, 96, 55, 78, 71
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OFFSET
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1,2
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COMMENTS
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This sequence is a self-inverse permutation of the positive integers that preserves the number of prime divisors (with or without multiplicity).
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LINKS
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Michael De Vlieger, Annotated log-log plot of a(n), n = 1..2^14, showing records in red, local minima in blue, highlighting primes in green, fixed points in gold, and composite prime powers in magenta.
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FORMULA
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a(prime(2*n)) = prime(2*n-1) (where prime(n) denotes the n-th prime number).
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EXAMPLE
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The first terms are:
n a(n) g=gcd(n, a(n)) n/g a(n)/g
-- ---- -------------- --- ------
1 1 1 1 1
2 3 1 2 3
3 2 1 3 2
4 6 2 2 3
5 7 1 5 7
6 4 2 3 2
7 5 1 7 5
8 12 4 2 3
9 15 3 3 5
10 14 2 5 7
11 13 1 11 13
12 8 4 3 2
13 11 1 13 11
14 10 2 7 5
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MATHEMATICA
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nn = 120; c[_] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == 0, AllTrue[{i/#, k/#}, PrimeQ] &@ GCD[i, k]], k++]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, 2, nn}]; Array[a, nn] (* Michael De Vlieger, May 22 2022 *)
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PROG
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(PARI) s=0; for (n=1, 67, for (v=1, oo, if (!bittest(s, v) && (n==1 || (isprime(n/g=gcd(n, v)) && isprime(v/g))), print1 (v", "); s+=2^v; break)))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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